AbstractChapter 7 explained the detection and hypothesis testing problems, Huffman codes and the situation where errors are independent and Gaussian. In this chapter, we prove the optimality of the Huffman code in Sect. 8.1 and the Neyman–Pearson Theorem in Sect. 8.2. Section 8.3 discusses the theory of jointly Gaussian random variables that is used to analyze the modulation schemes of Sect. 7.5 . Section 8.4 uses the results on jointly Gaussian random variables to explain hypothesis tests that arise when analyzing data. That section discusses the chi-squared test and the F-test. Section 8.5 is devoted to the LDPC codes that are widely used in high-speed communication links. These codes augment a group of bits to be transmitted over a noisy channel with additional bits computed from those in the group. When it receives the bits, when the augmented bits are not consistent, the receiver attempts to determine the bits that are most likely to have been corrupted by noise.